nLab
Berkovich space

Contents

Idea

Berkovich analytic spaces are a version of analytic spaces over nonarchimedean fields. Unlike the rigid analytic spaces (see rigid analytic geometry) of Tate, which are locally defined via maximal spectra of Tate algebras glued via rigidified Grothendieck topology, the Berkovich analytic spaces are defined with respect to the analytic spectrum, due to Vladimir Berkovich; this spectrum can be viewed as consisting of the data of prime ideal plus the extension of the norm to the residue field; thus the Berkovich spectrum has far more points (though fewer than, say, Huber’s adic spaces which may also contain valuations of higher order).

For more background see analytic geometry.

Definition of Berkovich analytic spaces

Let kk be a non-archimedean field.

Definition

Given nn \in \mathbb{N} and positive elements {r 1,,r nk}\{r_1, \cdots, r_n \in k\}, consider the sub-power series algebra over kk defined by

{1r 1T 1,,1r nT n}:={ νa νT νlim ν0a νr ν=0}. \{\frac{1}{r_1} T_1 , \cdots, \frac{1}{r_n}T_n\} := \left\{ \sum_\nu a_\nu T^\nu | \lim_{{\vert \nu\vert} \to 0} {\vert a_\nu \vert} r^\nu = 0 \right\} \,.

This is a commutative Banach algebra over kk with norm f=maxa νr ν{\Vert f \Vert} = max {\vert a_\nu\vert} r^\nu.

A kk-affinoid algebra is a commutative Banach kk-algebra AA for which there exists nn and {r i}\{r_i\} as above and an epimorphism

{1r 1T 1,,1r nT n}A \{\frac{1}{r_1} T_1 , \cdots, \frac{1}{r_n}T_n\} \to A

such that the norm on AA is the quotient norm.

If one can choose here r i=1r_i = 1 for all ii then AA is called strictly kk-affinoid.

The category of kk-affinoid spaces is the opposite category of the category of kk-affinoid algebras and bounded homomorphisms between them.

Via the analytic spectrum Spec anSpec_{an} there is a topological space associated with any kk-affinoid space. Often this underlying topological space is referred to as the analytic space.

Definition

An affinoid domain in an affoinoid space X=Spec anAX = Spec_{an} A is a closed subset VXV \subset X such that there is a homomorphism of kk-affinoid spaces

ϕ:Spec anA VX \phi : Spec_{an} A_V \to X

for some A VA_V, whose image is VV, and such that every other morphism of kk-affinoid spaces into XX whose image is contained in VV uniquely factors through this morphism.

Definition

A kk-analytic space is a topological space equipped with an atlas by affinoid domains.

See Berkovich 98, paragraph 2.

Properties

Local contractibility

A complex analytic manifold and a smooth complex analytic space is locally isomorphic to a polydisk and hence is trivially a locally contractible space. But over a non-archimedean field analytic spaces no longer need to be locally isomorphic to polydisks (but pp-adic polydisks are still contractible (Berkovich 90)). The following result establishes, under mild conditions, that general analytic spaces are nevertheless locally contractible.

Assume that the valuation on the ground field kk is nontrivial.

Definition

A kk-analytic space XX is called locally embeddable in a smooth space if each point of XX has an open neighbourhood isomorphic to a strictly kk-analytic domain in smooth kk-analytic space.

Theorem

Every kk-analytic space which is locally embeddable in a smooth space, def. 4, is a locally contractible space.

More precisely, every point of a locally smooth kk-analytic space has an open neighbourhood UU which is contractible, and which is a union U= i=1 U iU = \cup_{i = 1}^\infty U_i of analytic domains.

The local contractibility is Berkovich (1999), theorem 9.1. The refined statment in terms of inductive systems of analytic domains is in Berkovich (2004).

Examples

Applications

  • The proof of the local Langlands conjecture for GL nGL_n by Harris–Taylor uses étale cohomology on non-archimedean analytic spaces (in the sense of Berkovich) to construct the required Galois representations over local fields.

References

Introductions and reviews

Basic notions are listed in

  • M. Temkin, Non-archimedean analytic spaces (pdf slides)

A review of basic definitions and facts about affinoid and rigid kk-analytic spaces can be found in

  • Gaëtan Chenevier, lecture 5 (pdf)

See also the references at rigid analytic geometry.

A review of definitions and results on kk-analytic spaces is in

  • Vladimir Berkovich, pp-Adic analytic spaces, in Proceedings of the International Congress of Mathematicians, Berlin, August 1998, Doc. Math. J. DMV, Extra Volume ICM II (1998), 141-151 (pdf)

A more detailed set of lecture notes along these lines is

  • Vladimir Berkovich, Non-archimedean analytic spaces, lectures at the Advanced School on pp-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)

The Berkovich spectrum is discussed around section 2.1.4 in the course notes

A exposition of examples of Berkovits spectra is in

Original articles

  • Vladimir Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, (1990) 169 pp.

  • Vladimir Berkovich, Étale cohomology for non-Archimedean analytic spaces, Publ. Math. IHES 78 (1993), 5-161.

Discussion of local contractibility of smooth kk-analytic spaces is in

  • Vladimir Berkovich, Smooth pp-adic analytic spaces are locally contractible, Invent. Math. 137 1-84 (1999) (pdf)
  • Vladimir Berkovich, Smooth p-adic analytic spaces are locally contractible. II, in Geometric Aspects of Dwork Theory, Walter de Gruyter & Co., Berlin, (2004), 293-370. (pdf)

Relation to other topics

On the relation to buildings:

  • Annette Werner, Buildings and Berkovich Spaces (pdf)

Relation to integration theory

  • Vladimir Berkovich, Integration of 1-forms on pp-adic analytic spaces, Princeton University Press,

Aspects of the homotopy theory/étale homotopy of analytic spaces are discussed in

  • Aise Johan de Jong, Étale fundamental groups of non-archimedean analytic spaces, Mathematica, 97 no. 1-2 (1995), p. 89-118 (numdam)

Relation to formal schemes:

Discussion of Berkovich analytic geometry as algebraic geometry in the general sense of Bertrand Toën and Gabriele Vezzosi is in

  • Oren Ben-Bassat, Kobi Kremnizer, Non-Archimedean analytic geometry as relative algebraic geometry (arXiv:1312.0338)

Revised on December 4, 2013 02:08:47 by Urs Schreiber (82.113.99.55)